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Chaos. Solitons & Frorrals Vol. 6, pp. 171-201, 1995
Elsevicr Science Ltd
Rimed in Great Britain 0960-0779/95 $9.50 l .Ou
0960-0779(94)00259-2
Fractals in Biology and Medicine
S. HAVLIN,** S.V. BULDYREV, A.L. GOLDBERGER, RN. MANTEGNA,* S.M.
OSSADNIK; C.-K. PENG, M. SIMONSt8 and H.E. STANLEY
l Center for Polymer Studies and Department of Physics, Boston
University, Boston, MA USA
t Department of Physics, Bar-ilan University, Ramat-Gan,
Israel
t Cardiovascular Division, Harvard Medical School, Beth Israel
Hospital, Boston, MA 02215 USA
@ Department of Biology, Mfl Cambridge, MA 02139 USA
Abstract - Our purpose is to describe some recent progress in
applying fractal concepts to systems of relevance to biology and
medicine. We review several biological systems charac- terized by
fractal geometry, with a particular focus on the long-range
power-law correlations found recently in DNA sequences containing
noncoding material, Furthermore, we discuss the finding that the
exponent a quantifying these long-range correlations (fractal
complex- ity) is smaller for coding than for noncoding sequences.
We also discuss the application of fractal scaling analysis to the
dynamics of heartbeat regulation, and report the recent finding
that the normal heart is characterized by long-range
anticorrelations which are absent in the diseased heart.
1 Introduction
In the last decade it was realized that some biological systems
have no characteristic length or time scale, i.e., they have
fractal-or, more generally, self&fine-properties [ 1,2].
However, the fractal properties in different biological systems,
have quite different nature, origin, and appearance. In some cases,
it is the geometrical shape of a biological object itself that
exhibits obvious fractal features, while in other cases the fractal
properties are more hidden and can only be perceived if data are
studied as a function of time or mapped onto a graph in some
special way. After an appropriate mapping, such a graph may
resemble a mountain landscape, with jagged ridges of all length
scales from very small bumps to enormous peaks. Mathematically,
these landscapes can be quantified in terms of fractal concepts
such as self-affinity. The main part of the chapter is devoted to
the study of such hidden fractal properties that have been recently
discovered in DNA sequences and heartbeat activity.
2 Fractal Shapes
In contrast to compact objects, fractal objects have a very
large surface area. In fact, they are composed almost entirely of
surface. This observation explains why fractals are ubiquitous in
biology, where surface phenomena
0960-0779/94/.$07.00 @ 1994 Elsevier Science All rights reserved
SSDI
171
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172 S. HAVLIN et al.
are of crucial importance.
Lungs exemplify this feature (Fig. 1). The surface area of a
human lung is as large as a tennis court. The mam- malian lung is
made up of self-similar branches with many length scales, which is
the defining attribute of a fractal surface. The efficiency of the
lung is enhanced by this fractal property, since with each breath
oxygen and carbon dioxide have to be exchanged at the lung surface.
The structure of the bronchial tree has been quantita- tively
analyzed using fractal concepts [2,4], In particular, fractal
geometry could explain the power law decay of the average diameter
of the bronchial tube with the generation number, in contrast to
the classical model which predicts an exponential decay [6].
Not only the geometry of the respiratory tree is described by
fractal geometry, but also the time-dependent fea- tures of
insp~~ion. Specifically, Suki et al. [S] studied airway opening in
isolated dog lungs. During cons~t flow inflations, they found that
the lung volume changes in discrete jumps (Fig. l), and that the
probability distribu- tion function of the relative size z of the
jumps, II(z), and that of the time intervals t between these jumps,
II(t), follow a power law over nearly two decades of x and t with
exponents of 1.8 and 2.7, respectively. To interpret these fmclmgs,
they developed a branching airway model in which airways, labeled
ij, are closed with a uniform distribution of opening threshold
pressures P. When the airway opening pressure Pm exceeds PQ of an
airway, that airway opens along with one or both of its daughter
branches if Pij < P, for the daughters. Thus, the model predicts
avalanches of airway openings with a wide distribution of sixes,
and the statistics of the jumps agree with those II(z) and II(t)
measured ex~~men~ly. They concluded that power law dis~butiolls,
arising from avalanches triggered by threshold phenomena, govern
the recruitment of terminal airspaces.
A second example is the arterial system which delivers oxygen
and nutrients to all the cells of the body. For this purpose blood
vessels must have fractal properties [7,8]. The diameter dis~bution
of blood vessels ranging from capillaries to arteries follows a
power-law distribution which is one of the main characteristics of
fractals. Semetz et al. [9] have studied the branching patterns of
arterial kidney vessels. They analyzed the mass-radius relation and
found that it can be characterized by fractal geometry, with
fractal dimensions between 2.0 and 2.5. Sillily, the branching of
trees and other plants, as well as root systems have a fractal
nature [lo]. Moreover, the size distribution of plant-supported
insects was found to be related to the fractal distribution of the
leaves
Ull.
One of the most remarkable examples of a fractal object is the
surface of a cau~flower, where every little head is an almost exact
reduced copy of the whole head formed by intersecting Fibonacci
spirals of smaller heads, which in turn consist of spirals of
smaller and smaller heads, up to the fifth order of hierarchy (see
Fig. 8.0 in [3]). West and Goldberger were first to describe such a
Fibonacci fractal in the human lung [2].
Considerable interest in the biological community has also
arisen from the possibility that neuron shape can be quantified
using fractal concepts. For example, Smith et al. [ 121 studied the
fractal features of vertebrate central nervous system neurons in
culture and found that the fractal dimension is increased as the
neuron becomes more developed. Caserta et al. [ 13 J showed that
the shapes of qu~i-two-di~nsion~ retinal neurons can be character-
ized by a fractal dimension df. They found for fully developed
neurons in viva, df = 1.68 f 0.15, and suggest that the growth
mechanism for neurite outgrowth bears a direct analogy with the
growth model called difision limited aggregation (DLA). The
branching pattern of retinal vessels in a developed human eye is
also similar to DLA IS]. The fmctal dimension was estimated to be
about 1.7, in good agreement with DLA for the case of two
dimensions. For an alternative model for retinal growth see [
141.
The DLA-type model governing viscous fingering may also serve to
resolve the age-old paradox Why doesn r the ~~o~ck digest itself? [
151. Indeed, the concen~~ion of hydr~~o~c acid in the rn~~ stomach
after each meal is sufficient to digest the stomach itself, yet the
gastric epithelium normally remains undamaged in this harsh
environment. One protective factor is gastric mucus, a viscous
secretion of specialized cells, which forms a protective layer and
acts as a diffusion barrier to acid. Bicarbonate ion secreted by
the gastric epithelium is trapped in the mucus gel, estab~s~ng a
gradient from pH 1-2 at the lumen to pII 6-7 at the cell surface.
The
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Fractals in biology and medicine 173
Fig. 1. The dynamic mechanism responsible for filling the hmg
involves avalanches or bursts of air that occur in all
sizes-instead of an exponential distribution, one finds a power law
distribution [S]. The underlying cause of this scale-free
distributionof avalanches is the fact that every airway in the lung
has its own threshold below which it is not inflated. Shown here is
a diagram of the development of avalanches in the airways during
airway opening. At first. almost all airways whose threshold value
is smaller than the external pressure (red) are closed. Then the
airway opening pressure increases until a second threshold is
exceeded, and as a result all airways further up the tree whose
thresholds are smaller become inflated (green). The airway opening
pressure is successively increased until third, fourth, and fifth
thresholds are exceeded (yellow, brown, and blue). The last
threshold to be exceeded results in filling the airways colored
violet; we notice that this last avalanche opens up over 25% of the
total lung volume, thereby significantly increasing the total
surface area available for gas exchange. After [5]
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174 S. HAVLIN et al
Fig. 2. Photograph of a retinal neuron (nerve cell), the
morphology is similar to the DLA archetype. After [13]
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Fractals in biology and medicine 175
Fig. 3. Viscous fingers reflect the complex interface that
develops when one fluid is pumped through another of higher
viscosity. Shown is the formation of such viscous fingers or
channels when hydrochloric acid is injected into solutions of
gastric mucin. These channels may confine the acid and direct it to
the lumen, thus protecting the gastric mucosa from acidification
and ulceration; when the gastric glands contract, acid is ejected
under high enough pressure to form viscous fingers. After [
1.51
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176 S. HAVLIN et al
Fig. 4. A typical example of DLA-like colony patterns incubated
at 35OC for three weeks after inoculation on the surface of agar
plates containing initially 1 g/e of peptone as nutrient. This
pattern has a fractal dimension of {df g 1.72}2. After Matsushita
and Fujikawa [16]
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Fractals in biology and medicine
Fig. 5. Snapshots at successive times of the territory covered
by N random walkers for the case N = 500 for a sequence of times.
Note the roughening of the disc surface as time increases. The
roughening is characteristic of the experimental findings for the
diffusive spread of a population [21]. After [20], courtesy of P
Trunfio
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178 Fractals in biology and medicine
Fig. 8. The DNA walk representations of (a) human P-cardiac
myosin heavy chain gene sequence, showing the coding re- gions as
vertical golden bars. (b) the spliced together coding regions, and
(c) the bacte~ophage 1ambdaDNA which contains only coding regions.
Note the more complex fluctuations for (a) compared with thecoding
sequences (b) and(c). It is found that for almost all coding
sequences studied that there appear regions with one strand bias,
followed by regions of a differ- ent strand bias. fn this
presentation different step heights for purine and pyrimidine are
used in order to align the end point with the starting point. This
procedure is for graphicai display purposes only (to allow one to
visualizethe fluctuatious more easily) and is not used in any
analytic calculations. After (331
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Fractals in biology and medicine 179
0.6
a
0.5
0.4
30000 35000 40000 45000 50000
nucleotide position
55000 60000
Fig. 10. Analysis of section of Yeast Chromosome III using tbe
sliding box Coding Sequence Finder CSF algorithm. The value of the
long-range correlation exponent (Y is shown as a function of
position along the DNA chain. In this figure, the results for about
10% of the DNA are shown (from base pair #30,000 to base pair
#60,000). Shown as vertical bars are the putative genes and open
reading frames; denoted by the letter G are those genes that have
been more firmly identified (March 1993 version of GenBunk). Note
that the local value of (Y displays rninlrna where genes am
suspected, while between the genes o displays maxima. This behavior
corresponds to the fact that the DNA sequence of genes lacks
long-range correlations (a = 0.5 in Widealized limit), while the
DNA sequence in between genes possesses long-range iorrelations ((Y
x 0.6). After [57]
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180 S. HAVLIN et al.
Fig. 1 I. The DNA walk representation for the rat embryonic
skeletal myosin heavy chain gene (a = 0.63). At the top the entire
sequence is shown. In the middle the solid box shown in the top is
magnified. At the bottom the solid box shown in the middle is
magnified. The statistical self-similarity of these plots is
consistent with the existence of a scale-free or fractal phenomenon
which we call a fractal landscape. Note that one must magnify the
segment by different factors atong the E (horizontal) direction and
the y (vertical) direction; since F has the same units (dimension)
as y, these magnifica- tion factors Me and AfV (along e and y
directions respectively) are related to the scaling exponent Q: by
the simple relation cy = log(MU)/ log(ilft) [e.g., from top to
middle, log(M,)/ log(&) = log(2.07)/ log(3.2) = 0.631
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Fractals in biology and medicine 181
0 nucleotide distance, t
6000 0 nucleotide distance, e
6000
Fig. 12. The DNA walk representations of (a) 8 cDNA sequences
from the MHC family and (b) the corresponding genes. DNA landscapes
are plotted so that the end points have the same vertical
displacement as the starting points [33]. The graphs are for yeast,
amoeba, worms: C. efegans, Brugiu makzyi, drosophila, chicken, rat
and human (from top to bottom, left to right). The shaded areas in
(b) denote coding regions of the genes. The DNA walks for the genes
show increasing complexity with evolution. In contrast, the cDNA
walks all show remarkably similar crossover patterns due to
sequential up-hill and downhill slopes representing different
purine/pyrimidine strand biases in the regions coding for the head
and tail of the MHC molecule, respectively. After [I%]
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182 S. HAVLN et al.
ii E O 7500 0 9000 z 10 drosoDhila chicken 5 600 600 ._ .a
., 0 23000 0' 23000
rat human
24000 0 23000 nucleotide distance, t nucleotide distance, t
Fig. 12b.
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Fractals in biology and medicine 183
puzzle, then, is how hydrochloric acid, secreted at the base of
gastric glands by specialized parietal cells, tra- verses the mucus
layer to reach the lumen without acidifying the mucus layer.
Bhaskar et al. [ 151 resolved this puzzle by experiments that
demonstrate the the possibility that flow of hydrochloric acid
through mucus involves viscous fingering-the phenomenon that occurs
when a fluid of lower viscosity is injected into a more viscous one
(see Fig. 3). Specifically, Bhaskar et al. demonstrated that
injection of hydrochloric acid through solutions of pig gastric
mucin produces fingering patterns which are strongly dependent on
pH, mucin concentration, and acid flow rate. Above pH 4, discrete
fingers are observed, while below pH 4. hydrochloric acid neither
penetrates the mucin solution nor forms fingers. These in vitro
results suggest that hydrochloric acid secreted by the gastric
gland can penetrate the mucus gel layer (pH 5-7) through narrow
fingers, whereas hydrochloric acid in the lu- men (pH 2) is
prevented from diffusing back to the epithelium by the high
viscosity of gastric mucus gel on the luminal side.
Yet another example of DLA-type growth is bacterial colony
spread on the surface of agar (gel with nutrient) plates [ 161 (see
Fig. 4). Vicsek et al. [ 171 studied bacterial colony growth on a
strip geometry which results in a self-afline surface (see Fig.
13.19 in [ 181). They calculated the roughness exponent cr for this
surface and found LY = 0.78 f 0.07. The interfacial pattern
formation of the growth of bacterial colonies was studied
systematically by Ben-Jacob et al. [ 191. They demonstrated that
bacterial colonies can develop a pattern similar to morpholo- gies
in diffusion-limited growth observed in solidification and
electro-chemical deposition. These include fractal growth,
dense-branching growth, compact growth, dendritic growth and chiral
growth. The results indicate that the interplay between the micro
level (individual bacterium) and the macro level (the colony) play
a major role in selecting the observed morphologies similar to
those found in nonliving systems.
Another example of fractal interface appears in ecology, in the
problem of the territory covered by N diffusing particles [20], see
Fig. 5. As seen from the figure, the territory initially grows with
the shape of a disk with a relatively smooth surface until it
reaches a certain size, at which point the surface becomes
increasingly rough. This phenomenon may have been observed by
Skellam [21] who plotted contours delineating the advance of the
muskrat population and noted that initially the contours were
smooth but at later times they became rough (see Fig. 1 in
[21]).
other biological contexts in which fractal scaling seems to be
relevant are the relation between brain size and body weight [22],
between bone diameter and bone length [23], between muscle force
and muscle mass [23], and between an organisms size and its rate of
producing energy and consuming food [24].
3 Long-Range Power Law Correlations
In recent years long-range power-law correlations have been
discovered in a remarkably wide variety of systems. Such long-range
power-law correlations are a physical fact that in turn gives rise
to the increasingly appreciated fractal geometry of nature [
1,3,18,25-281. So if fractals are indeed so widespread, it makes
sense to anticipate that long-range power-law correlations may be
similarly widespread. Indeed, recognizing the ubiquity of long-
range power-law correlations can help us in our efforts to
understand nature, since as soon as we find power-law correlations
we can quantify them with a critical exponent. Quantification of
this kind of scaling behavior for apparently unrelated systems
allows us to recognize similarities between different systems,
leading to underlying unifications that might otherwise have gone
unnoticed.
Usually correlations decay exponentially, but there is one major
exception: at the critical point [29], the expo- nential decay of
(la) turns into to a power law decay
c, N (l/+-s+?
Many systems drive themselves spontaneously toward critical
points [30.3 11. One of the simplest models exhibit-
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lS4 s. HAVLIN et al
ing such self-organized criticality* is invasion percolation, a
generic model that has recently found applicability to describing
anomalous behavior of rough interfaces. Instead of occupying all
sites with random numbers below a pre-set parameter p, in invasion
percolation one grows the incipient infinite cluster right at the
percolation threshold by the trick of occupying always the
perimeterlitgwhose random number is smallest. Thus small clus- ters
are certainly not scale-invariant and in fact contain sites with a
wide distribution of random numbers. As the mass of the cluster
increases, the cluster becomes closer and closer to being scale
invariant or fractal. Such a system is said to drive itself to a
self-organized critical state [32].
In the fohowing sections we will attempt to summarize the key
findings of some recent work [33-571 suggesting that-under suitable
conditions-the sequence of base pairs or nucleotides in DNA also
displays power-law correlations. The underlying basis of such power
law correlations is not understood at present, but this discovery
has intriguing implications for molecular evolution and DNA
structure, as well as potential practical applications for
distinguishing coding and noncoding regions in long nucleotide
chains.
4 Information Coding in DNA
The role of genomic DNA sequences in coding for protein
structure is well known [58,59]. The human genome contains
information for approximately 100,000 different proteins, which
define all inheritable features of an individual. The genomic
sequence is likely the most sophisticated information database
created by nature through the dynamic process of evolution. Equally
remarkable is the precise transformation of information
(duplication, decoding, etc) that occurs in a relatively short time
interval.
The building blocks for coding this information are called
nucleotides. Each nucleotide contains a phosphate group, a
deoxyribose sugar moiety and either a puke or a pyrimidine base.
lXvo purines and two pyrimidines are found in DNA. The two purines
are adenine (A) and guanine (G); the two pyrimidines are cytosine
(C) and thymine (T). The nucleotides are linked end to end, by
chemical bonds from the phosphate group of one nu- cleotide to the
deoxyribose sugar group of the adjacent nucleotide, forming a long
polymer (polynucleotide) chain. The information content is encoded
in the sequential order of the bases on this chain. Therefore, as
far as the information content is concerned, a DNA sequence can be
most simply represented as a symbolic sequence of four letters: A,
C, G and T, as shown in Fig. 6.
Fig. 6. The base pairing of two double helix DNA strands. The
two chains of black pentagons and circles represent sugar-phosphate
backbones of DNA strands linked by the hydrogen bonds (dashed
lines) between complementary base pairs
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Fractals in biology and medicine 185
In the genomes of high eukaryotic organisms only a small portion
of the total genome length is used for protein coding (as low as 5%
in the human genome). For example, genes are separated from each
other by intergenic sequences which are not used for coding
proteins and which (especially in mammalian genomes) can be several
times longer than genes, Furthermore, in 1977 it was discovered
that genes themselves have inclusions which are not used for coding
proteins. A gene is transcripted to RNA (pre-mRNA) and then some
segments of the pm- mRNA are spliced out during the formation of
the smaller mRNA molecule. The mRNA then serves as the template for
assembling protein. The segments of the chromosomal DNA that are
spliced out during the formation of a mature mRNA are called
introns (for intervening sequences). The coding sequences are
called exuns (for expressive sequences).
The role of introns and intergenomic sequences constituting
large portions of the genome remains unknown. Furthermore, only a
few quantitative methods are currently available for analyzing
information which is possibly encrypted in the noncoding part of
the genome.
5 Conventional Statistical Analysis of DNA Sequences
DNA sequences have been analyzed using a variety of models that
can basically be considered in two categories. The first types are
local analyses; they take into account the fact that DNA sequences
are produced in sequen- tial order; therefore, the neighboring
nucleotides will affect the next attaching nucleotide. This type of
analysis, represented by n-step Markov models, can indeed describe
some observed short-range correlations in DNA se- quences. The
second category of analyses is more global in nature; they
concentrate on the presence of repeated patterns (such as periodic
repeats and interspersed base sequence repeats) that are chiefly
found in eukaryotic genomic sequences. A typical example of
analysis in this category is the Fourier transform, which can
identify repeats of certain segments of the same length in
nucleotide sequences [58].
However, DNA sequences are more complicated than these two
standard types of analysis can describe. There- fore it is crucial
to develop new tools for analysis with a view toward uncovering the
mechanisms used to code other types of information. Promising
techniques for genome studies may be derived from other fields of
scien- tific research, including time-series analysis, statistical
mechanics, fractal geometry, and even linguistics.
6 The DNA Walk
One interesting question that may be asked by statistical
physicists would be whether the sequence of the nu- clcotides
A,C,G, and T behaves like a one-dimensional ideal gas, where the
fluctuations of density of certain particles obey Gaussian law, or
if them exist long range correlations in nucleotide content (as in
the vicinity of
6 -
CAATTCTGTTCTGGTAATCC DNA sequence
Fig. 7. Schematic illustration showing the definition of the DNA
walk
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186 S. HAVLM et ul.
a critical point). These result in domains of all size with
different nucleotide concentrations. Such domains of various sizes
were known for a long time but their origin and statistical
properties remain unexplained. A natural language to describe
heterogeneous DNA structure is long-range correlation analysis,
borrowed from the theory of critical phenomena [29].
6.1 Graphical Representation
In order to study the scale-invariant long-range correlations of
a DNA sequence, we first introduced a graphical representation of
DNA sequences, which we term afractal landscape or DNA walk [33].
For the conventional one-dimensional random walk model [60,61], a
walker moves either up [u(i) = +l] or down [u(i) = -11 one unit
length for each step i of the walk. For the case of an uncorrelated
walk, the direction of each step is independent of the previous
steps. For the case of a correlated random walk, the direction of
each step depends on the history (memory) of the walker
[62-64].
One definition of the DNA walk is that the walker steps up [u(i)
= +l] if a pyrimidine (C or T) occurs at position i along the DNA
chain, while the walker steps down [u(i) = --I] if a purine (A or
G) occurs at po- sition i (see Fig. 7). The question we asked was
whether such a walk displays only short-range correlations (as in
an n-step Markov chain) or long-range correlations (as in critical
phenomena and other scale-free fractal phenomena).
There have also been attempts to map DNA sequence onto
multi-dimensional DNA walks [34,65]. However, recent work [57]
indicates that the original purine-pyrimidine rule provides the
most robust results, probably due to the purine-pyrimidine chemical
complementarity.
The DNA walk allows one to visualize directly the fluctuations
of the purine-pyrimidine content in DNA se- quences: Positive
slopes on Fig. 8 correspond to high concentration of pyrimidines,
while negative slopes corre- spond to high concentration of
purines. Visual observation of DNA walks suggests that the coding
sequences and intron-containing noncoding sequences have quite
different landscapes. Figure 8a shows a typical example of a gene
that contains a significant fraction of base pairs that do not code
for amino acids. Figure 8b shows the DNA walk for a sequence formed
by splicing together the coding regions of the DNA sequence of this
same gene (i.e., the cDNA). Figure 8c displays the DNA walk for a
typical sequence with only coding regions. Landscapes for
intron-containing sequences show very jagged contours which consist
of patches of all length scales, reminiscent of the disordered
state of matter near critical point. On the other hand, coding
sequences typically consist of a few lengthy regions of different
strand bias, resembling domains in the system iv the ferromagnet
state. These observations can be tested by rigorous statistical
analysis. Figure 8 naturally motivates a quantification of these
fluctuations by calculating the net displacement of the walker
after C steps, which is the sum of the unit steps u(i) for each
step i. Thus y(C) E & u(i).
6.2 Correlations and Fluctuations
An important statistical quantity characterizing any walk
[60,61] is the root mean square fluctuation F(C) about the average
of the displacement; F(e) is defined in terms of the difference
between the average of the square and the square of the
average,
- ~ P(e) = [Ay(e) - Ay(l)] = [AY@)]" - ay(e>, (2)
of a quantity Ay(e) defined by Ay(e) E y(& + l) - y(&,)
(see also Chaps. 1 and 5). Here the bars indicate an
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Fractals in biology and medicine 187
average over all positions la in the gene. Operationally, this
is equivalent to (a) using calipers preset for a fixed distance e,
(b) moving the beginning point sequentially from.!, = 1 to & =
2, - . a and(c) calculating the quantity Ay(e) (and its square) for
each value of &, and (d) averaging all of the calculated
quantities to obtain F(e).
The mean square fluctuation is related to the auto-correlation
function
C(e) = u(&)u(& + e) - $&$ through the relation F2(e)
= & 2 c(j - i). (3) i=l j=i
The. calculation of F(C) can distinguish three possible types of
behavior.
(9
(W
(iii)
If the base pair sequence were random, then C(e) would be zero
on average [except C( 0) = 11, so F(e) N Cl/ (as expected for a
normal random walk). If there were local correlations extending up
to a characteristic range R (such as in Markov chains), then C(e) N
exp(-e/R); nonetheless the asymptotic (.! > R) behavior F(e) N
!?I2 would be unchangedfrom the purely random case. If there is no
characteristic length (i.e., if the correlation were
infinite-range), then the scaling property of C(e) would not be
exponential, but would most likely to be a power law function, and
the fluctuations will also be described by a power law
with ff # l/2.
(4)
Figure 8a shows a typical example of a gene that contains a
significant fraction of base pairs that do not code for amino
acids. It is immediately apparent that the DNA walk has an
extremely jagged contour which corresponds to long-range
correlations. Figure 9 shows double logarithmic plots of the mean
square fluctuation function F(P) as a function of the linear
distance L along the DNA chain for a typical intron-containiig
gene.
The fact that the data for intron-containing and intergenic
(i.e. noncoding) sequences are linear on this double logarithmic
plot confirms that F(e) N P. A least-squares fit produces a
straight line with slope cy substantially larger than the
prediction for an uncorrelated walk, a = l/2, thus providing direct
experimental evidence for the presence of long-range
correlations.
On the other hand, the dependence of F(e) for codiigsequences is
not linear on the log-log plot: its slope un- dergoes a crossover
from 0.5 for small C to 1 for large 4. However, if a single patch
is analyzed separately, the log-log plot of F(E) is again a
straight line with the slope close to 0.5. This suggests that
within a large patch the coding sequence is almost
uncorrelated.
It is known that functional proteins usually form a single
compact three-dimensional conformation that corre- sponds to the
global energy minimum in the conformational space. Recently,
Shakhnovich and Gutin [66] found that in order to have such a
minimum it is sufficient that an amino acid sequence forms au
uncorrelated random sequence. The finding of Peng et al. 1331 of
the lack of long range correlations in the coding nucleotide
sequences provides more evidence for this hypothesis, since there
exist almost one-to-one correspondence between amino acid sequences
and theii nucleotide codes, Furthermore, this finding may also
indicate. that the lack of long range correlations in the amino
acid sequences is, in fact, a necessary condition for a functional
biologically active pro- tein.
7 Differences Between Correlation Properties of Coding and
Noncoding Regions
The initial report [33] on long-range (scale-invariant)
correlations only in noncoding DNA sequences has gener-
-
188 S. HAVLIN er al.
cb) Omg. b 0.8-
Fig. 9. (a) Double logarithmic plots of the mean square
fluctuation function F(e) as a function of the linear distance e
along the DNA chain for the rat embryonic skeletal myosin heavy
chain gene (0) and its intron-spliced sequence (0). (b) The
corresponding local slopes, crlocd, based on pairs of successive
data points of part (a). We see that the values of QI are roughly
constant. For this specific gene, the sequence with exons removed
has an even broader scaling regime than the DNA se quence of the
entire gene, indicated by the fact that part (a) is linear up to
10,000 nucleotides. After [42]
ated contradicting responses. Some [34,35,38,39] support our
initial finding, while some [35,40,44,50] disagree. However, the
conclusions of Refs. [36] and [35,40,44,50] are inconsistent with
one anorkr in that [35] and [50] doubt the existence of long-range
correlations (even in noncoding sequences) while [36] and [40,44]
conclude that even coding regions display long-range correlations
(a > l/2). Prabhu and Claverie [40] claim that their analysis of
the putative coding regions of the yeast chromosome III [67]
produces a wide range of exponent vaf- ues, some larger than 0.5.
The source of these contradicting claims may arise from the fact
that, in addition to normal statistical fluctuations expected for
analysis of rather short sequences, coding regions typically
consist of only a few lengthy regions of alternating strand bias.
Hence conventional scaling analyses cannot be applied reliably to
the entire sequence but only to sub-sequences.
Peng et al. [56] have recently applied the bridge method to DNA,
and have also developed a similar method
-
Fractals in biology and medicine 189
specifically adapted to handle problems associated with
non-stationary sequences which they term detnmfed fructuution
analysis (DFA).
The idea of the DFA method is to compute the dependence of the
standard error of a linear interpolation of a DNA walk F#) on th e
size of the interpolation segment C. The method takes into account
differences in local nucleotide content and may be applied to the
entire sequence which has lengthy patches. In contrast with the
orig- inal F(e) function, which has spurious crossovers even for!
much smaller than a typical patch size, the de&ended function
Fd(.tY) shows linear behavior on the log-log plot for all length
scales up to the characteristic patch size, which is of the order
of a thousand nucleotides in the coding sequences. For .!? close to
the characteristic patch size the log-log plot of Fd(l) has an
abrupt change in its slope.
The DFA method clearly supports the difference between coding
and noncoding sequences, showing that the coding sequences are less
correlated than noncoding sequences for the length scales less than
1000, which is close to characteristic patch size in the coding
regions. One source of this difference is the tandem repeats
(sequences such as AAAAAA. . .). which are quite frequent in
noncoding sequences and absent in the coding sequences.
To provide an unbiased test of the thesis that noncoding regions
possess but coding regions lack long-range cor- relations, Gssadnik
et al. [57] analyzed several artificial uncorrelated and correlated
control sequences of size lo5 nucleotides using the GRAIL neural
net algorithm [68]. The GRAIL algorithm identified about 60
putative exons in the uncorrelated sequences, but only about 5
putative exons in the correlated sequences.
Using the DFA method, we can measure the local value of the
correlation exponent a: along the sequence (see Fig 10) and find
that the local minima of a as a function of a nucleotide position
usually correspond to noncoding regions, while the local maxima
correspond to noncoding regions. Statistical analysis using the DFA
technique of the nucleotide sequence data for yeast chromosome III
(315,338 nucleotides) shows that the probability that the observed
correspondence between the positions of minima and coding regions
is due to random coincidence is less than 0.0014. Thus, this
method-which we called the coding sequence finder (CSF)
algorithm-can be used for finding coding regions in the newly
sequenced DNA, a potentially important application of DNA walk
analysis.
8 Long-Range Correlations and Evolution
What is the biological meaning of the finding of long-range
correlations in DNA? If two nucleotides whose posi- tions differ by
1000 base pairs were uncorrelated, then there might be no meaning.
However, the finding that they are correlated suggests some
underlying organizational property. The long-range correlations in
DNA sequences are of interest because they may be an indirect clue
to its three-dimensional structure [45,54] or a reflection of
certain scale-invariant properties of long polymer chains [53,55].
In any case, the statistically meaningful long- range
scale-invariant (see Fig. 11) correlations in noncoding regions and
their absence in coding regions will need to be accounted for by
future explanations of global properties in gene organization and
evolution.
Molecular evolutionary relationships are usually inferred from
comparison of coding sequences, conservation of intron/exon
structure of related sequences, analysis of nucleotide
substitutions, and construction of phylogenetic trees 1691. The
changes observed are conventionally interpreted with respect to
nucleotide sequence composition (mutations, deletions,
substitutions, alternative splicing, transpositions, etc.) rather
than overall genomic organi- zation.
Very recently, Buldyrev et al. [55] sought to assess the utility
of DNA correlation analysis as a complementary method of studying
gene evolution. In particular, they studied the changes in fractal
complexity of nucleotide organization of a single gene family with
evolution. A recent study by Voss [36] reported that the
correlation ex-
-
190 s. HAVLIN et al.
ponent derived from Fourier analysis was lowest for sequences
from organelles, but paradoxically higher for in- vertebrates than
vertebrates. However, this analysis must be interpreted with
caution since it was based on pooled data from different gene
families rather than from the quantitative examination of any
single gene family (see also [70,71]).
The hypothesis that the fractal complexity of genes from higher
animals is greater than that of lower animals, using single gene
family analysis was tested in [55]. This analysis focuses on the
genome sequences from the conventional (Type II) myosin heavy chain
(MI-K) family. Such a choice limits potential bias that may arise
sec- ondary to non-uniform evolutionary pressures and differences
in nucleotide content between unrelated genes. The MHC gene family
was chosen because of the availability of completely sequenced
genes from a phyloge- netically diverse group of organisms, and the
fact that their relatively long sequences are well-suited to
statistical analysis.
The landscape produced by DNA walk analysis reveals that each
MHC cDNA consists of two roughly equal parts with significant
differences in nucleotide content (Fig. 12). The first part that
codes for the heavy meromyosin or head of the protein molecule has
a slight excess of purines (52% purines and 48% pyrimidines); the
second part that codes for the light meromyosin or tail has about
63% purines and 37% pyrimidines. The absolute nucleotide contents
are not shown in the graphical representation of Fig. 12a because
we subtract the average slope from the landscape to make relative
fluctuations around the average more visible. Indeed, one can
easily see from Fig. 12a that the relative concentration of
pyrimidines in the first part (uphill region) of the myosin cDNA is
much higher than in the second (downhill region).
The landscapes of Fig. 12 show that the coding sequences of
myosins remain practically unchanged with evolu- tion, while the
entire gene sequences become more heterogeneous and complex. The
quantitative measurements of the exponent o by DFA method confirm
this visual observation showing that for all coding sequences of
MIX family Q: M 0.5. In contrast, for entire genes of MHC family,
the value of (Y monotonically increases from lower eukaryotes to
invertebrates and from invertebrates to vertebrates [55]. A
stochastic model of random deletions and insertions of DNA portions
was developed in [55] to explain this finding; see also
[72-771.
Two major theories have been advanced to explain the origin and
evolution of introns. One suggests that precursor genes consisted
entirely of coding sequences and introns were inserted later in the
course of evolution to help facilitate development of new
structures in response to selective pressure, perhaps, by means of
exon shuffling [78]. The alternative theory suggests that precursor
genes were highly segmented and subsequently organisms not
requiring extensive adaptation or new development or, perhaps,
facing the high energetic costs of replicating unnecessary
sequences, lost their introns [79,80]. Support for these hypotheses
has remained largely conjectural; no models have been brought
forward to supporl either process. The landscape analysis of the
MIX gene family and the stochastic model [53,55] here are more
consistent with the former view.
9 Other Biological Systems with Long-Range Correlations
The catalog of systems in which power law correlations appear
has grown rapidly in recent years [32,81,82]; see also Chap. 2.
What do we anticipate for other biological systems? Generally
speaking, when entropy wins over energy- i.e., randomness dominates
the behavior-we find power laws and scale invariance. The absence
of characteristic length (or time) scales may confer important
biological advantages, related to adaptability of response 121.
Biological systems sometimes are described in language that makes
one think of a Swiss watch. Such mechanistic or Rube Goldberg
descriptions must in some sense be incomplete, since it is only
some appropriately-chosen averages that appear to behave in a
regular fashion. The trajectory of each individual bio- logical
molecule is of necessity random -albeit correlated. Thus one might
hope that recent advances in under- standing correlated randomness
[83,62-64] could be relevant to biological phenomena.
-
Fractals in biology and medicine 191
9.1 The Human Heartbeat
Traditionally, clinicians describe the normal electrical
activity of the heart as regular sinus rhythm. However, cardiac
interbeat intervals fluctuate in a complex, apparently erratic
manner in healthy subjects even at rest. Anal- ysis of heart rate
variability has focused primarily on short time oscillations
associated with breathing (0.15-0.40 Hz) and blood pressure control
(- 0.1 Hz) [go]. Fourier analysis of longer heart rate data sets
from healthy in- dividuals typically reveals a l/f-like spectrum
for frequencies < 0.1 Hz [84-87].
Peng et al. [88] recently studied scale-invariant properties of
the human heartbeat time series, the output of a complicated
integrative control system. The analysis is based on the digitized
electrocardiograms of beat-to-beat heart rate fluctuations over
very long time intervals (up to 24 h x lo5 beats) recorded with an
ambulatory monitor. The time series obtained by plotting the
sequential intervals between beat n and beat n+ 1, denoted by B(n),
typ- ically reveals a complex type of variability. The mechanism
underlying such fluctuations is related to competing neuroautonomic
inputs. Parasympathetic (vagal) stimulation decreases the firing
rate of pacemaker cells in the hearts sinus node; sympathetic
stimulation has the opposite effect. The nonlinear interaction
(competition) be- tween these two branches of the involuntary
nervous system is the postulated mechanism for much of the erratic
heart rate variability recorded in healthy subjects, although
non-autonomic factors may also be important.
To study these dynamics over large time scales, the time series
is passed through a digital filter that removes fluctuations of
frequencies > 0.005 beat-, and plot the result, denoted by
BL(~), in Fig. 13. One observes a more complex pattern of
fluctuations for a representative healthy adult (Fig. 13a) compared
to the smoother pattern of interbeat intervals for a subject with
severe heart disease (Fig. 13b). These heartbeat time series
produce a contour reminiscent of the irregular landscapes that have
been widely studied in physical systems.
To quantitatively characterize such a landscape, Peng et al.
introduce a mean fluctuation function F(n), defined as
F(n) = IBL(n' + n) - Bt(n')l,
where the bar denotes an average over all values of n. Since
F(n) measures the average difference between two interbeat
intervals separated by a time lag n, F(n) quantifies the magnitude
of the fluctuation over different time scales n.
Figure 13c is a log-log plot of F(n) vs n for the data in Figs.
13a and 13b. This plot is approximately linear over a broad
physiologically-relevant time scale (200 - 4000 beats) implying
that
F(n) N nQ. (6)
It is found that the scaling exponent (Y is markedly different
for the healthy and diseased states: for the healthy heartbeat data
a is close to 0, while a is close to 0.5 for the diseased case.
Note that (Y = 0.5 corresponds to a random walk (a Brownian
motion), thus the low-frequency heartbeat fluctuations for a
diseased state can be interpreted as a stochastic process, in which
the heartbeat intervals I(n) = B(n + 1) - B(n) are uncorrelated for
n 2 200.
To investigate these dynamical differences, it is helpful to
study further the correlation properties of the time series. It is
useful to study I(n) because it is the appropriate variable for the
aforementioned reason. Since I(n) is stationary, one can apply
standard spectral analysis techniques [3] (see also Chap. 1).
Figures 14a and 14b show the power spectra S,(f), the square of the
Fourier transform amplitudes for I(n), derived from the same data
sets (without filtering) used in Fig. 13. The fact that the log-log
plot of Sr(f) vs f is linear implies
-
192 S. HAVLM et al.
I t I 1 2 3 4 5 3
Beat Number [x104]
05. 0 1 2 3 4 5 6
Beat Number [x10*]
0.01
c a = 0 Normal
4
lo3 n [beat]
10
Fig. 13. The interbeat interval &(n) after low-pass
filtering for (a) a healthy subject and (b) a patient with severe
cardiac disease (dilated c~iomyopa~y). The healthy heartbeat time
series shows more complex ~uc~~io~ compared to the dis- eased heart
rate fluctuation pattern that is close to random walk (brown)
noise. (c) Log-log plot of F(n) vs n. The circles represent F(n)
calculated from data in (a) and the triangles from data in (b). The
two best-fit lines have slope (Y = 0.07 and CY = 0.49 (fit from 200
to 4CBO beats). The two lines with slopes LY =: 0 and Q = 0.5
correspond to l/f noise and brown noise, respectivdy. We observe
that F(n) saturates for large n (of the order of 5000 beats),
because the heartbeat interval are subjected to physiological
constraints that camrot be arbitrarily large or small. The low-pass
filter removes all Fourier components for f 2 fC. The results shown
here correspond to fC = 0.005 beat-, but similar findings are
obtained for other choices of fC < 0.005. This cut-off frequency
fC is selected to remove components of heart rate variability
associated with physiologic respiration or pathologic Cheyne-Stokes
breathing as well as oscillations associated with baroretlex
activation (Mayer waves). After [SS]
-
Fractals in biology and medicine
SI(f) N $.
193
(7)
The exponent /3 is related to Q! by @ = 2a - 1[62]. Furthermore,
@ can serve as aa indicator of the presence and type of
correlations:
(i) If ,B = 0, there is a0 correktioa ia the time series I(n)
(white noise),
f (beaf)
f (beaf)
Fig. 14. The power spectmm Sr(ff for the interbeat interval
incremeat sequences over N 24 hours for the same subjects ia Fig.
13. (a) Data from a healthy adult. The best-tit line for the low
frequency region has a slope p = -0.93. The heart rate spectrum is
plotted as a function of inverse best number (beat-l) rather than
fnquency (time-i) to obviate the need to interpolate data points.
The spectral data an smoothed by averaging over 50 values. (b) Data
from a patieat with severe heart failure. The best-fit line has
slope 0.14 for the low frequency region, f c fc = 0.005 beat-*. Ike
appearance of a pathologic, characteristic time scale is associated
with a spectral peak (arrow) at about 10q2 beat- (correspoadiag to
Cheyne-stokes respiratk& After [88]
-
194 S. HAVLIN et al.
Fig. 15. Musical mapping of two heartbeat times series, derived
from normal (top) and pathologic (bottom) data sets. The original
heart beat time series were obtained from 24 hour recordings
consisting of about IO6 heartbeats. The heartbeat time series were
then low-pass filtered to remove fluctuations > 0.05 (beat-),
roughly equivalent to averaging every 200 beats. The pattern of
fluctuations in the normal is more complex than that of the music
generated from the abnormal data sets. Musical compositions based
on these times series are available on cassette by request along
with the scores; contact Zachary D. Goldberger (e-mail: ary at
astro.bih.harvard.edu). There is a nominal charge for copying and
mailing. After [97J, courtesy of Z.D. Goldberger
-
Fractals in biology and medicine 195
(ii) If 0 < p < 1, then I(n) is correlated such that
positive values of I are likely to be close (in time) to each
other,
and the same is true for negative I values. (iii) If -1 < p
< 0, then I(n) is also correlated; however, the values of I are
organized such that positive and
negative values are more likely to alternate in time
(anti-correlation) [62].
For the diseased data set, we observe a flat spectrum (p GZ 0)
in the low frequency region (Fig. 14b) contirm- ing that r(n) are
not correlated over long time scales (low frequencies). Therefore,
I(n), the first derivative of B(n), can be interpreted as being
analogous to the velocity of a random walker, which is uncorrelated
on long time scales, while B(n)-corresponding to the position of
the random walker-are correlated. However, this correlation is of a
trivial nature since it is simply due to the summation of
uncorrelated random variables.
In contrast, for the data set from the healthy subject (Fig.
14a), we obtain p 5z - 1, indicating nontrivial long- range
correlations in B(n)-these correlations are not the consequence of
summation over random variables or artifacts of non-stationarity.
Furthermore, the anti-correlation properties of I(n) indicated by
the negative p value are consistent with a nonlinear feedback
system that kicks the heart rate away from extremes. This tendency,
however, does not only operate on a beat-to-beat basis (local
effect) but on a wide range of time scales. To our knowledge, this
is the first explicit description of long-range anticorrelations in
a fundamental biological variable, namely the interbeat interval
increments.
Furthermore, the histogram for the heartbeat intervals
increments is found to be well-described by a L&y stable
distribution. For a group of subjects with severe heart disease, it
is found that the distribution is unchanged, but the long-range
correlations vanish. Therefore, the different scaling behavior in
health and disease must relate to the underlying dynamics of the
heartbeat. Applications of this analysis may lead to new
diagnostics for patients at high risk of cardiac disease and sudden
death.
9.2 Physiological Implications
The finding of nontrivial long-range correlations in healthy
heart rate dynamics is consistent with the observation of
long-range correlations in other biological systems that do not
have a characteristic scale of time or length. Such behavior may be
adaptive for at least two reasons. (i) The long-range correlations
serve as an organizing principle for highly complex, nonlinear
processes that generate fluctuations on a wide range of time
scales. (ii) The lack of a characteristic scale helps prevent
excessive mode-locking that would restrict the functional respon-
siveness of the organism. Support for these related conjectures is
provided by observations from severe diseased states such as heart
failure where the breakdown of long-range correlations is often
accompanied by the emer- gence of a dominant frequency mode (e.g.,
the Cheyne-Stokes frequency). Analogous transitions to highly peri-
odic regimes have been observed in a wide range of other disease
states including certain malignancies, sudden cardiac death,
epilepsy and fetal distress syndromes.
The complete breakdown of normal long-range correlations in any
physiological system could theoretically lead to three possible
diseased states: (i) a random walk (brown noise), (ii) highly
periodic behavior, or (iii) com- pletely uncorrelated behavior
(white noise). Cases (i) and (ii) both indicate only trivial
long-range correlations of the types observed in severe heart
failure. Case (iii) may cotrespond to certain cardiac arrhythmias
such as fibrillation. More subtle or intermittent degradation of
long-range cormlation properties may provide an early warning of
incipient pathology. Finally, we note that the long-range
correlations present in the healthy heartbeat indicate that the
neuroautonomic control mechanism actually drives the system away
from a single steady state. Therefore, the classical theory of
homeostasis, according to which stable physiological processes seek
to main- tain constancy [89], should be extended to account for
this dynamical, far from equilibrium, behavior.
-
196 S. HAVLIN et al.
9.3 Human Writings
Long-range correlations have been found recentty in human
writings [8 11. A novel, a piece of music or a com- puter program
can be regarded as a one-dimensional string of symbols. These
strings can be mapped to a one- dimensional random walk model
similar to the DNA walk (Sect. 6) allowing calculation of the
correlation ex- ponent a using (4a). Values of Q between 0.6 and
0.9 were found for various texts.
An interesting fractal feature of languages was found in 1949 by
Zipf [90]. He observed that the frequency of words as a function of
the word order decays as a power law (with a power close to - 1)
for more than four orders of magnitude. A theory for this empirical
finding, based on assumptions of coding words in the brain, was
given by ~andelbrot [1,91]. A related interesting statistical
measure of short-range correlations in languages and in general
series sequences is the entropy and redundancy defined by Shannon
1921.
Ions, such as potassium and sodium, cannot cross the lipid cell
membrane. They can, however, enter or exit the celi through ion
channel proteins that are distributed on the cell membr~e. These
proteins spon~eously fluc- tuate between open or closed states.
Liebovitch f933 found that the histograms of the open and closed
duration times of some channels are self-similar and behave as
power laws. This approach may provide new models for the ion
channel gating mechanisms.
9.5 Fractal Music and the Heartbeat
Fourier analysis of instantaneous fluctuations in amplitude as
well as inter-note intervals for certain classical music pieces
(e.g., Bachs First Brandenburg Concerto) reveals a l/f distribution
over a broad frequency range [94-961. Voss and Clarke [95] used an
algorithm for generating l/f-noise to compose music.
Based on the observation of different scaling patterns for
healthy and pathologic heartbeat time series [88], it was very
recently postulated [97] that (i) actual biological rhythms such as
the heartbeat might serve as a more natu- ral template for musical
compositions than artificially-generated noises, and (ii) audibly
appreciable differences between the note series of healthy and
diseased hearts could potentially serve as the basis for a
clinically useful diagnostic test. Accordingly, a computer program
was devised to map heart rate fluctuations onto intervals of the
diatonic musical scale [97]. As anticipated, the nor&
(l/f-like) heartbeat obtained from the low pass fil- tered time
series reported in [SS] generated a more variable (complex) type of
music than that generated by the abnormal times series (Fig.
15).
The musicality of these transcriptions is intriguing and
supports speculations about the brains possible role as a
translator/manipulator of biological l/f-like noise into
aesthetically pleasing art works. Current investigations are aimed
at extending these preliminary observations by (i) comparing the
musicality of note sequences gen- erated by natural (biologic~) vs.
artificial (computer simulated) correlated and uncorrelated noises,
and (ii) using heartbeat time series as the template for
simultaneously generating fluctuations in musical rhythm and
intensity, not only pitch.
9.6 Fractal Approach to Biological Evolution
Fossil data suggest that evolution of biological species takes
place as inte~i~ent bursts of activity, separated
-
Fractals in biology and medicine 197
by relatively long periods of quiescence [98]. Recently Bak and
Sneppen [99] suggested that these spontaneous catastrophic
extinctions may be related to the power law distribution of
avalanches of growth observed in a model of self-organized
criticality (SOC). Such SOC models are reminiscent of recent
surface growth models based on the concept of directed invasion
percolation [ 1001.
Acknowledgements
We are grateful to R. Bansil, A.-L. Barabasi, K.R. Bhaskar, E
Caserta, G. Daccord, W. Eldred, P Garik, Z.D. Goldberger, Z.
Hantos, J.M. Hausdorff, R.E. Hausman, P Ivanov, T.J. LaMont. H.
Larralde, M.E. Matsa, J. Mi- etus, A. Neer, J. Nittmann, E P&k.
I. Rabin, F. Sciortino, A. Shehter, M.H.R. Stanley, B. Suki, P.
Trunlio, M. Uirleja, and G.H. Weiss for major contributionrj to
those results reviewed here that represent. collaborative re-
search efforts. We also wish to thank C. Cantor, C. DeLisi, M.
Frank-Kamenetskii, A.Yu. Grosberg. G. Huber, I. Labat, L.
Liebovitch, G.S. Michaels, P. Munson, R. Nossal. R. Nussinov, R.D.
Rosenberg, J.J. Schwartz, M. Schwartz, E.I. Shakhnovich, M.F.
Shlesinger, N. Shworak. and E.N. Trifonov for valuable discussions.
Partial support was provided to to SVB and HES by the National
Science Foundation, to ALG by the G. Harold and Leila Y. Mathers
Charitable Foundation, the National Heart, Lung and Blood Institute
and the National Aeronautics and Space Administration, to SH by
Israel-USA Binational Science foundation, and to C-KP by an NIWNIMH
Postdoctoral NRSA Fellowship.
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